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150 XIII INTEGRATION 15. Let / be a lower (or upper) semicontinuous real-valued function on Rn. Let X be a locally compact space and ui9..., un universally measurable finite real-valued functions. Show that/(#i,..., un) is universally measurable (cf. Problem 17). 16. With the notation of Section 12.7, Problem 3, show that there exist n universally measurable mappings t\>zk(t) of C" into C (1 ^ k <[ n) such that Pt(X) = X" + hX"-1 + + * - fl (X ~ **(0). (By induction on the degree Ğ, using Problem 15.) 17. Let / be the continuous function on I = [0, 1 ] which was defined in Section 4.2, Problem 2(d), such that the restriction of/to the Cantor set K is a bijection of K onto I. Let g be the function defined by g(x) = x +/(jc); then g is a homeomorphism of I onto 21 = [0,2], such that g(K) is a compact set of measure 1 (with respect to Lebesgue measure), although K is negligible. If h(x) = g~1(2x), then h is a homeo- morphism of I onto I such that h~^(K^ is not negligible with respect to Lebesgue measure. Moreover, if A is a nonmeasurable set (with respect to Lebesgue measure) contained in #(K), then the set B = /z(J-A) is negligible and therefore measurable. The function <pB ° h is not measurable, although h is continuous and <pB is measurable (cf. (16.23)). 18. (a) Let/be a finite continuous real-valued function on R. Show that there exists a real-valued function g on R which is universally measurable and such that/(#(*)) = x for all x e/(R) (cf. Section 12.7, Problem 1). (b) Assume that there exists a partition of I == [0,1] into a family (Hn)neZ of non- measurable sets (with respect to Lebesgue measure) such that Card (Hn) = Card(R) for all neZ. Let K be the Cantor set (Section 4.2, Problem 2) and F = (J (K + Ğ), n e Z so that F is a negligible closed set. Show that there exists a bijection/: R-^R such that/(F n ]n, n + 1 [) = Hn for all n, and such that / is linear in each of the com- ponent intervals of OF- Then / is measurable with respect to Lebesgue measure, but f~l is not. 19. Let X be a locally compact space and ft a positive measure on X. Let D be a de- numerable dense subset of R. Show that a mapping/: X-ğR is ft-measurable if and only if, for each r E D, the set of points x e X such that/(x) ^ r is ft-measurable. 20. Suppose ft bounded and ft(X) = 1. If /^ 0 is an integrable function, show that (1 +/2)1/2 is integrable and that, if A = (fdp,, then (1 + A2)1/2 g J(l +/2)1/2 dfji g 1 + A. (Use Problem 14(a) of Section 13.8.) 21. Let X, Y, Z be three locally compact spaces, if /: X -> Y and g : Y -+ Z are universally measurable, show that g of is universally measurable. (To show that# °/ is/^-measur- able where ^ is any positive measure on X, reduce to the case where X is compact and consider the measure/(/x) on Y (Section 13.4, Problem 8).) In particular, if B <= Y is universally measurable, then/-1(B) is universally measurable.